### Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

### Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

### Fermat's Poser

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

##### Age 14 to 16 Short Challenge Level:

The diagram on the right shows a square with side length $2\text{m}$ and lines drawn to its sides from its centre $O$. The points $A$, $B$, $C$ and $D$ are all on different sides of the square.

The lines $OA$ and $OB$ are perpendicular, as are $OC$ and $OD$.